APLIMAT - JOURNAL OF APPLIED MATHEMATICS
VOLUME 5 (2012), NUMBER 3
APLIMAT - JOURNAL OF APPLIED MATHEMATICS
VOLUME 5 (2012), NUMBER 3
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APLIMAT - JOURNAL OF APPLIED MATHEMATICS VOLUME 5 (2012), NUMBER 3
ALGEBRA AND ITS APPLICATIONS
JASEM Milan: A CAUCHY COMPLETION OF DUALLY RESIDUATED 15 LATTICE ORDERED SEMIGROUPS SEIBERT Jaroslav: ON THE HYPER-WIENER INDEX AND 25 POLYNOMIAL OF A GRAPH. VANŽUROVÁ Alena : HEXAGONAL AND GOLDEN QUASIGROUPS 31 VANŽUROVÁ Alena, BARTOŠKOVÁ Zuzana: TOYODA'S 41 THEOREM AND QUASIGROUP MODES
APLIMAT - JOURNAL OF APPLIED MATHEMATICS VOLUME 5 (2012), NUMBER 3
GEOMETRY AND ITS APPLICATIONS
ARSAN Güler Gürpınar, ÇİVİ Gülçin: GEODESIC MAPPINGS 51 PRESERVING THE EINSTEIN TENSOR OF WEYL SPACES BASTL Bohumír, LÁVIČKA Miroslav: SMOOTH CURVES 57 APPROXIMATION BY CHORD-LENGTH CURVES BIZZARRI Michal, LÁVIČKA Miroslav: ON COMPUTING 67 APPROXIMATE PARAMETERIZATIONS OF ALGEBRAIC SURFACES ÇIVI Gülçin , ARSAN Güler Gürpınar: ON HOLOMORPHICALLY 79 PROJECTIVE MAPPINGS OF KAHLER- WEYL SPACES CHEPURNA Olena, MIKEŠ Josef: HOLOMORPHICALLY PROJECTIVE MAPPINGS OF HYPERBOLICALLY KAHLER SPACES 85 PRESERVING THE EINSTEIN TENSOR CHODOROVÁ Marie, CHUDÁ Hana, SHIHA Mohsen: ON COMPOSITION OF CONFORMAL AND HOLOMORPHICALLY 91 PROJECTIVE MAPPINGS BETWEEN CONFORMALLY K¨AHLERIAN SPACES JUKL Marek, JUKLOVÁ Lenka, MIKEŠ Josef: MULTIPLE 97 COVARIANT DERIVATIVE AND DECOMPOSITION PROBLEMS KAŇKA Miloš, ELIÁŠOVÁ Lada: THE CURVATURES OF SPECIAL 105 FUNCTIONS IN ECONOMY PETRASOVA Alena, CZANNER Silvester, CHALMERS Alan, WOLKE Dieter: THE UTILITY OF THE VIRTUAL REALITY IN DEEPER 113 UNDERSTANDING OF PEOPLE'S EXPERIENCES OF INFANT FEEDING SLABÁ Kristýna, BASTL Bohumír: CIRCLE-PRESERVING 123 SUBDIVISION SCHEME BASED ON APOLLONIUS' CIRCLE VANŽUROVÁ Alena, JUKL Marek: PARALLELOGRAM SPACES 133 AND MEDIAL QUASIGROUPS VANŽUROVÁ Alena, PIRKLOVÁ Petra: METRIZABLE CONNECTIONS AND RESTRICTIVELY VARIATIONAL CONNECTIONS 141 IN AFFINE MANIFOLDS VELICHOVÁ Daniela: MULTIDIMENSIONAL MANIFOLDS AS 151 MINKOWSKI OPERATION PRODUCTS VOICU Nicoleta: FINSLERIAN CONNECTIONS AND THE EQUATIONS 159 OF SPINNING CHARGED PARTICLES IN GENERAL RELATIVITY
APLIMAT - JOURNAL OF APPLIED MATHEMATICS VOLUME 5 (2012), NUMBER 3
STATISTICAL METHODS IN TECHNICAL AND ECONOMIC SCIENCES AND PRACTICE
ANDRADE Marina, FERREIRA Manuel Alberto M.: CRIME SCENE INVESTIGATION THROUGH DNA TRACES USING BAYESIAN 167 NETWORKS ANDRADE, Marina, FERREIRA, Manuel Alberto M.: CIVIL AND 173 CRIMINAL IDENTIFICATION WITH BAYESIAN NETWORKS ARSHINOVA Tatyana: RISK MANAGEMENT OF EQUITY PORTFOLIO CONSTRUCTION ON THE BASIS OF DATA 185 ENVELOPMENT ANALYSIS APPROACH BARTOŠOVÁ Jitka, FORBELSKÁ Marie: GMM MODEL OF AT- RISK-OF-POVERTY CZECH HOUSEHOLDS DEPENDING ON THE AGE 195 AND SEX OF THE HOUSEHOLDER (EU-SILC 2005-2009) BEZRUCKO Aleksandrs: LATVIAN GDP: TIME SERIES 205 FORECASTING USING VECTOR AUTO REGRESSION FERREIRA Manuel Alberto M., ANDRADE Marina: A METHOD TO APPROXIMATE FIRST PASSAGE TIMES DISTRIBUTIONS IN DIRECT 217 TIME MARKOV PROCESSES FERREIRA, Manuel Alberto M., ANDRADE, Marina: SOJOURN 225 TIMES IN JACKSON NETWORKS FJODOROVS Jegors, MATVEJEVS Andrejs: COPULA BASED 241 SEMIPARAMETRIC REGRESSIVE MODELS HECKENBERGEROVÁ Jana, MAREK Jaroslav, SOUČKOVÁ Jitka, TUČEK Pavel: NONSMOOTH FUNCTION APPROXIMATION IN 249 PRACTICAL CHANGE POINT PROBLEM JAROŠOVÁ Eva: COMPARISON OF TWO BAYESIAN APPROACHES 259 TO SPC MALÁ Ivana: QUANTILE CHARACTERISTICS OF CONDITIONAL 269 DISTRIBUTIONS OF FINITE MIXTURES MISKOLCZI Martina, LANGHAMROVÁ Jitka: MULTISTATE LIFE TABLES: APPLICATION OF THE METHOD ON THE MARRIAGE 279 CAREER MOŠNA František: TWO APPLICATIONS OF PROBABILITY IN THE 287 THEORY OF RELIABILITY AND MAINTENANCE NEUBAUER Jiří: INFLATION MODELING AND COINTEGRATION 293 ŽIŽKA David: APPLICATION OF RELEVANCE VECTOR MACHINE TO 301 FORECASTING VOLATILITY IN CZECH FINANCIAL TIME SERIES
LIST OF REVIEWERS
Andrade Marina, Professor Auxiliar University Institute of Lisbon, Lisboa, Portugal
Bartošová Jitka, RNDr., PhD University of Economics, Jindřichův Hradec, Czech Republic
Baštinec Jaromír, doc. RNDr., CSc. FEEC, Brno University of Technology, Brno, Czech Republic
Beránek Jaroslav, doc. RNDr., CSc. Masaryk University, Brno, Czech Republic
Biswas Md. Haider Ali, Associate Engineering and Technology School, Khulna Professor University, Belize
Bittnerová Daniela, RNDr., CSc. Technical Univerzity of Liberec, Liberec, Czech Republic
Brabec Marek, Ing., PhD Academy of Sciences of the Czech Republic, Praha, Czech Republic
Buikis Maris, Prof. Dr. Riga Technical University, Riga, Latvia
Cyhelský Lubomír, Prof. Ing., DrSc. Vysoká škola finanční a správní, Praha, Czech Republic
Dorociaková Božena, RNDr., PhD University of Žilina, Žilina, Slovak Republic
Emanovský Petr, Doc. RNDr., PhD Palacky University, Olomouc, Czech Republic
Ferreira Manuel Alberto M., Professor University Institute of Lisbon, Lisboa, Portugal Catedrático
Filipe José António, Professor Auxiliar IBS - IUL, ISCTE - IUL, Lisboa , Portugal
Habiballa Hashim, RNDr. PaedDr., University of Ostrava, Ostrava, Czech Republic PhD
Habiballa Hashim, RNDr. PaedDr., University of Ostrava, Ostrava, Czech Republic PhD
Hošková-Mayerová Šárka, doc. University of Defence, Brno, Czech Republic RNDr., PhD
11
Hošpesová Alena, doc. PhDr., PhD Jihočeská univerzita, České Budějovice, Czech Republic
Iorfida Vincenzo Lamezia Terme, Italy
Iveta Stankovičová, PhD UK Bratislava, Bratislava, Slovak Republic
Jancarik Antonin, PhD Charles University, Prague, Czech Republic
Jukl Marek, RNDr., PhD Palacky University, Olomouc, Czech Republic
Kráľ Pavol, RNDr., PhD Matej Bel University, Banska Bystrica, Slovak Republic
Kunderová Pavla, doc. RNDr., CSc. Palacky University, Olomouc, Czech Republic
Kvasz Ladislav, Prof. Charles University, Prague, Czech Republic
Langhamrová Jitka, doc. Ing., CSc University of Economic in Prague, Prague Czech Republic
Linda Bohdan, doc. RNDr., CSc. University of Pardubice, Pardubice, Czech Republic
Maroš Bohumil, doc. RNDr., CSc. University of Technology, Brno, Czech Republic
Matvejevs Andrejs, DrSc., Ing. Riga Technical university, Riga, Latvia
Mikeš Josef, Prof. RNDr., DrSc. Palacky University, Olomouc, Czech Republic
Milerová Helena, Bc. Charles University, Prague, Czech Republic
Miroslav Husek Charles University, Prague, Czech Republic
Miskolczi Martina, Mgr., Ing. University of Economics in Prague, Prague, Czech Republic
Morkisz Paweł, AGH University of Science and Technology, Krakow, Poland
Mošna František, RNDr., PhD Czech Univ. of Life Sciences, Praha, Czech Republic
Paláček Radomir, RNDr., PhD VŠB - Technical University of Ostrava, Ostrava, Czech Republic
Pospíšil Jiří, Prof. Ing., CSc. Czech Technical University of Prague, Prague, Czech Republic
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Potůček Radovan, RNDr., PhD University of Defence, Brno, Czech Republic
Radova Jarmila, doc. RNDr., PhD University of Economics, Prague, Czech Republic
Rus Ioan A., Professor Babes-Bolyai University of Cluj-Napoca, Cluj-Napoca, Romania
Růžičková Miroslava, doc. RNDr., University of Žilina, Žilina, Slovak Republic CSc.
Segeth Karel, Prof. RNDr., CSc. Academy of Sciences of the Czech Republic, Prague , Czech Republic
Slaby Antonin, Prof. RNDr., PhDr., University of Hradec Kralove, Hradec Kralove, Czech CSc. Republic
Sousa Cristina Alexandra, Master Universidade Portucalense Infante D. Henrique, Porto, Portugal
Svoboda Zdeněk, RNDr., CSc. FEEC, Brno University of Technology, Brno, Czech Republic
Šamšula Pavel, doc. PaedDr., CSc Charles University, Prague, Czech Republic
Torre Matteo, Laurea in Matematica Scula Secondaria Superiore, Alessandria, Italy
Trojovsky Pavel, RNDr., PhD University of Hradec Kralove, Hradec Kralove, Czech Republic
Trokanová Katarína, Doc. Slovak Technical University, Bratislava, Slovak Republic
Ulrychová Eva, RNDr. University of Finance and Administration, Prague, Czech Republic
Vanžurová Alena, doc. RNDr., CSc. Palacký University, Olomouc, Czech Republic
Velichová Daniela, doc. RNDr., CSc. Slovak University of Technology, Bratislava, Slovak mim.prof. Republic
Vítovec Jiří, Mgr., PhD Brno University of Technology, Brno, Czech Republic
Voicu Nicoleta, Dr. Transilvania University of Brasov, Romania, Brasov, Romania
Volna Eva, doc. RNDr. PaedDr. PhD University of Ostrava, Ostrava, Czech Republic
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Wimmer Gejza, Professor Slovak Academy of Sciences, Bratislava, Slovak Republic
Zeithamer Tomáš R., Ing., PhD University of Economics, Prague, Czech Republic
14
A CAUCHY COMPLETION OF DUALLY RESIDUATED LATTICE ORDERED SEMIGROUPS JASEM Milan, (SK)
Abstract. In this paper convergence with a fixed regulator in dually residuated lattice ordered semigroups is investigated and a u-Cauchy completion of a strong dually residuated lattice ordered semigroup is constructed. It is also shown that this completion is uniquely determined up to isomorphism.
Key words and phrases. u-uniform convergence, Cauchy sequence, dually residuated lattice ordered semigroups.
Mathematics Subject Classification. Primary 06F05.
1 Introduction
Dually residuated lattice ordered semigroups (DRl-semigroups) were introduced and studied by Swamy in [14], [15], [16]. DRl-semigroups were also investigated by Kov´aˇr [10], [11], K¨uhr [12] and by the author [8]. Galatos and Tsinakis [7] proved that DRl-semigroups are equivalent to commutative GBL-algebras. Birkhoff [1] and Luxemburg and Zaanen [13] studied relatively uniform convergence of se- quences in vector lattices. Relatively uniform convergence in lattice ordered groups was dealt with by Cern´ˇ ak and Lihov´a[5]andCern´ˇ ak and Jakub´ık [6]. Convergence with a fixed regulator in lattice ordered groups was studied by Cern´ˇ ak [2], [3] and Cern´ˇ ak and Lihov´a[4]. This paper is a continuation of the paper [9] where convergence with a fixed regulator in DRl-semigroups was introduced and studied. In the present paper Cauchy sequences are investigated and a u-Cauchy completion of a strong DRl-semigroup has been constructed.
2 Preliminaries
We review some notions and notations used in the paper. Aplimat - Journal of Applied Mathematics
Asystem A =(A, +, ≤, −) is called a dually residuated lattice ordered semigroup if and only if (1) (A, +, ≤) is a commmutative lattice ordered semigroup with zero element 0, i. e. (A, +) is a commutative semigroup with zero 0 and (A, ≤) is a lattice with lattice operations ∧ and ∨ such that a +(b ∨ c)=(a + b) ∨ (a + c)anda +(b ∧ c)=(a + b) ∧ (a + c), (2) given a, b in A there exists a least x in A such that b + x ≥ a, and this x is denoted by a − b, (3) (a − b) ∨ 0+b ≤ a ∨ b for all a, b ∈ A, (4) (a − a) ≤ 0 for each a ∈ A. Any DRl-semigroup can be equationally defined as an algebra with the binary operations +, ∨, ∧, −, by replacing (2) by the equations: x +(y − x) ≥ y, x − y ≤ (x ∨ z) − y, (x + y) − y ≤ x [14, Theorem 1]. For any a and b in a DRl-semigroup A we shall write |a − b| =(a − b) ∨ (b − a)(|a − b| is called the symetric difference of a and b. ) This notation arising from lattice ordered groups is different from one used by Swamy in [14], however it is most suitable for our case. The symetric difference satisfies the following conditions: (i) |a − b|≥0, |a − b| =0ifandonlyifa = b, (ii) |a − b| = |b − a|, (iii) |a − c|≤|a − b| + |b − c|. Any DRl-semigroup is an autometrized algebra with the symetric difference [14, Theorem 9]. We use N for the set of all positive integers. Let A be a DRl-semigroup. We denote A+ = {x ∈ A; x ≥ 0}. An element x ∈ A+ is said to be Archimedean if whenever y ∈ A+ and ny ≤ x for each n ∈ N, then y =0. A DRl-semigroup A is called strong if x, y ∈ A and 2x ≤ 2y implies x ≤ y. Any abelian lattice ordered group is a strong DRl-semigroup and hence any Archimedean l-group is also a strong DRl-semigroup. We shall need the following propositions from [14]. Let A be a DRl-semigroup, a, b, c ∈ A. Then (P1) a ≤ b if and only if a − b ≤ 0 (Lemma 7), (P2) a ≤ b implies (b − a)+a = b (Lemma 8), (P3) a ≤ b implies a − c ≤ b − c and c − b ≤ c − a (Lemma 3), (P4)(a ∨ b) − c =(a − c) ∨ (b − c) (Lemma 4). Luxemburg and Zaanen [13] introduced notions of a u-uniform convergence and of a relatively uniform convergence of sequences for vector lattices and Cern´ˇ ak and Lihov´a [5] for lattice ordered groups. Analogous definition of a u-uniform convergence we shall use for DRl-semigroups.
+ Definition 2.1 Let A be a DRl-semigroup, (xn) asequenceinA, u ∈ A . It is said that a u sequence (xn) in A converges u-uniformly to an element x ∈ A, written xn → x, if the following condition is satisfied: (C3)foreachk ∈ N there exists nk ∈ N, such that k|xn − x|≤u for each n ∈ N,n≥ nk.
The element u in the Definition 2.1 is called a convergence regulator. u If xn → x, we say that x is a u−limit of (xn).
16 volume 5 (2012), number 3 Aplimat - Journal of Applied Mathematics
If we take the same regulator for all sequences we get convergence which is called convergence with a fixed regulator. Cern´ˇ ak and Lihov´a have shown that if convergence regulator u in lattice ordered group is not Archimedean then a sequence can have more u-limits. So, it is convenient to have an Archimedean element in the role of convergence regulator. Basic properties of the convergence with a fixed regulator in DRl-semigroups were estab- lished in [9]. It was shown that if convergence regulator u in a strong DRl-semigroup B is an Archimedean element, then u-limits are uniquely determined (Theorem 1) and if (xn), (yn) are sequences in u u u u u B and xn → x and yn → y, then xn + yn → x + y, xn − yn → x − y, xn ∨ yn → x ∨ y, u xn ∧ yn → x ∧ y (Theorem 2).
3 Cauchy sequences and a u-Cauchy completion
+ Definition 3.1 Let B be a DRl-semigroup, u ∈ B . Asequence(xn) in B is called a u-Cauchy sequence, if for each k ∈ N there exists nk ∈ N such that u ≥ k|xm − xn| for each m, n ∈ N, m, n ≥ nk. Throughout the rest of the paper A will be a strong DRl-semigroup and an Archimedean element u of A will be a fixed regulator for all sequences in A.
In [9] it was showed that each u-convergent sequence in A is a u-Cauchy sequence (Theorem 6) and that if (xn)and(yn) are u-Cauchy sequences, then (xn + yn), (xn − yn), (xn ∨ yn), (xn ∧ yn) are u-Cauchy sequences, too (Theorem 7). Let C be the set of all u-Cauchy sequences in A. Let (xn), (yn) ∈ C. We put (xn)+(yn)=(xn + yn). Further we set (xn) ≤ (yn) if and only if xn ≤ yn for each n ∈ N. If (xn), (yn) ∈ C and 2(xn) ≤ 2(yn), then (xn) ≤ (yn) [9, Theorem 2(v)].
We denote by (x) the sequence (x,x,x,...)inA. Clearly, the u−limit of this sequence is x. In [9] it was also proved that (C, +, ≤) is a strong DRl-semigroup with zero (0) and lattice op- erations ∨ and ∧ such that (xn)∨(yn)=(xn ∨yn), (xn)∧(yn)=(xn ∧yn) for all (xn), (yn) ∈ C. Further, (xn) − (yn)=(xn − yn) for all (xn), (yn) ∈ C.
Swamy [16, p. 71] defined an ideal and a convex sub-DRl-semigroup of a DRl-semigroup as follows. Definition 3.2 A non-empty subset I of A is called an ideal of A if and only if: (i) a, b ∈ I implies a + b ∈ I, (ii) a ∈ I, b ∈ A and |b − 0|≤|a − 0| imply b ∈ I.
Definition 3.3 Non-empty subset S of A is called a convex sub-DRl-semigroup if and only if the following conditions are satisfied: (i) if a, b ∈ S, then a + b, a − b, a ∧ b, a ∨ b ∈ S, (ii) if a, b ∈ S, x ∈ A and a ∧ b ≤ x ≤ a ∨ b, then x ∈ S.
volume 5 (2012), number 3 17 Aplimat - Journal of Applied Mathematics
u The set E of all sequences (xn)inA such that xn → 0 is an ideal of C and a convex sub-DRl- semigroup of C. [9, Theorems 9 and 10 ] By a congruence relation on a DRl-semigroup B we mean an equivalence relation, having the substitution property with respect to all the operations: +, ∨, ∧ and −. Swamy [16, Theorem 1.2] showed that ideals of any DRl-semigroup correspond one to one to its congruence relations. Further, he showed that if I is an ideal of B, then the binary relation ϑ(I) defined by (a, b) ∈ ϑ(I) if and only if |a − b|∈I is a congruence relation on B [16, p. 72]. Hence, the factor semigroup A∗ = C/ϑ(E) is a DRl-semigroup. ∗ We denote by (xn) the congruence class modulo ϑ(E) containing the sequence (xn). Re- ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ call that (xn) +(yn) =(xn + yn) , (xn) − (yn) =(xn − yn) , (xn) ∧ (yn) =(xn ∧ yn) , ∗ ∗ ∗ ∗ ∗ ∗ (xn) ∨ (yn) =(xn ∨ yn) , for all (xn) , (yn) ∈ A .
Lemma 3.4 (i) E =(0)∗. ∗ (ii) If (xn) ∈ C, (xn) ∈ (xn) , then (xn) − (xn), (xn) − (xn) ∈ E.
Proof. (i) Let (xn) ∈ E. Since E is a DRl-semigroup, we have |(xn) − (0)|∈E. Hence ∗ ∗ (xn) ∈ (0) and thus E ⊆ (0) . ∗ Let (yn) ∈ (0) . Then |(yn) − (0)|∈E. Since |(yn) − (0)| = ||(yn) − (0)|−(0)|∈E, we have ∗ (yn) ∈ E. Hence (0) ⊆ E. ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ (ii) If (xn) ∈ (xn) , then (xn) =(xn) . Thus (xn − xn) =(xn) − (xn) =(xn) − (xn) = ∗ ∗ (xn − xn) =(0) = E. Hence (xn) − (xn) ∈ E. Analogously, (xn) − (xn) ∈ E.
Lemma 3.5 Let (xn), (yn) ∈ C. Then the following conditions are equivalent: ∗ ∗ (i) (xn) ≤ (yn) , (ii) (xn) ≤ (yn)+(tn) for some (tn) ∈ E, (tn) ≥ (0), ∗ ∗ (iii) for each (xn) ∈ (xn) there exists (yn) ∈ (yn) such that (xn) ≤ (yn).
∗ ∗ ∗ ∗ ∗ Proof. (i)⇔(ii) Clearly, (xn) ≤ (yn) iff (xn) ∨ (yn) =(yn) iff ((xn) ∨ (yn)) − (yn)= ∗ ∗ |((xn) ∨ (yn)) − (yn)|∈E. Let (zn)=((xn) ∨ (yn)) − (yn). Thus (zn) ≥ (0). If (xn) ≤ (yn) , then (zn) ∈ E. In view of (P2)weobtain(xn) ≤ (xn) ∨ (yn) = (((xn) ∨ (yn)) − (yn)) + (yn)= (zn)+(yn)=(yn)+(zn). Conversely, if (xn) ≤ (yn)+(zn), where (0) ≤ (zn) ∈ E, then from (P3) it follows that (xn) − (yn) ≤ ((zn)+(yn)) − (yn) ≤ (zn). Accordingto(P4), ((xn) ∨ (yn)) − (yn)=((xn) − (yn)) ∨ (0) ≤ (zn) ∨ (0) = (zn) ∈ E. The equivalence of (i) and (iii) is obvious because ϑ(E) is a lattice congruence.
Lemma 3.6 A∗ is a strong DRl-semigroup.
∗ ∗ ∗ ∗ ∗ ∗ ∗ Proof. Let (xn) ,(yn) ∈ A ,2(xn) ≤ 2(yn) . Then (2xn) ≤ (2yn) . By Lemma 3.5, (2xn) ≤ (2yn)+(tn) ≤ (2yn)+(2tn), where (tn) ∈ E,(0) ≤ (tn). This implies 2xn ≤ 2yn +2tn =2(yn +tn) for each n ∈ N. Then xn ≤ yn + tn for each n ∈ N. This implies (xn) ≤ (yn)+(tn). By Lemma ∗ ∗ ∗ 3.5, (xn) ≤ (yn) . Hence DRl-semigroup A is strong.
18 volume 5 (2012), number 3 Aplimat - Journal of Applied Mathematics
Lemma 3.7 (u)∗ is an Archimedean element in A∗.
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ Proof. Let (xn) ∈ A , (0) ≤ (xn) , m(xn) ≤ (u) for each m ∈ N. From (0) ≤ (xn) and ∗ Lemma 3.5 it follows that (0) ≤ (xn) for some (xn) ∈ (xn) . Thus 0 ≤ xn for each n ∈ N, ∗ ∗ ∗ ∗ ∗ ∗ (xn) =(xn) . Then (mxn) = m(xn) = m(xn) ≤ (u) for each m ∈ N. By Lemma 3.5, u (mxn) ≤ (u)+(tn) for some (tn) ∈ E,(tn) ≥ (0). Since tn → 0 for each k ∈ N there exists nk ∈ N such that tn = |tn − 0|≤k|tn − 0|≤u for each n ∈ N,n≥ nk. Thus for each m ∈ N we have mxn ≤ u + tn ≤ 2u, where n ∈ N,n≥ nk. Hence 2pxn ≤ 2u for each p ∈ N. Then pxn ≤ u u for each p ∈ N. This yields xn = 0 for each n ∈ N,n ≥ nk. Thus xn → 0 and hence (xn) ∈ E. ∗ ∗ ∗ ∗ Then (xn) =(xn) = E. Therefore (u) is an Archimedean element in A .
∗ u ∗ (u) ∗ Lemma 3.8 Let (xn) be a sequence in A. If xn → 0, then (xn) → (0) .
u Proof. Let (xn) be a sequence in A, xn → 0. Hence for each k ∈ N there exists nk ∈ N such that k|xn − 0|≤u for each n ∈ N,n≥ nk. Then (u) ≥ (k|xn − 0|)=k(|xn − 0|)=k|(xn) − (0)| ∗ ∗ ∗ ∗ ∗ for each n ∈ N,n≥ nk. Thus (u) ≥ k|(xn) − (0) | for each n ∈ N,n≥ nk. Hence (xn) →(0) .
Lemma 3.9 Let (xn) ∈ C, l ∈ N, a1,...,al ∈ A.Letyn = xl+n−1 for each n ∈ N i. e. (yn)=(xl,xl+1,xl+2,...).Letz1 = a1,...,zl = al, zn = xn for each n ∈ N,n≥ l +1 i. e. (zn)=(a1,...,al,xl+1,xl+2,...).Then (i) (yn), (zn) ∈ C, ∗ ∗ ∗ (ii) (xn) =(yn) =(zn) .
Proof. Since (xn) ∈ C, for each k ∈ N there exists nk ∈ N such that k|xm − xn|≤u for each m, n ∈ N,m,n≥ nk. (i) Since l + m − 1 ≥ nk,l+ n − 1 ≥ nk, we have u ≥ k|xl+m−1 − xl+n−1| = k|ym − yn| for each m, n ∈ N,m,n≥ nk. Therefore (yn) ∈ C. Further, if we take nk ∈ N,nk ≥ l, then we have k|zm − zn| = k|xm − xn|≤u for each m, n ∈ N,m,n≥ nk. Hence (zn) ∈ C. (ii) If we take m = l + n − 1, we have u ≥ k|xl+n−1 − xn| = k|yn − xn| = k||yn − xn|−0| for u ∗ ∗ each n ∈ N,n≥ nk. Thus |yn − xn| → 0. Hence |(yn) − (xn)|∈E. This implies (xn) =(yn) . u ∗ ∗ Clearly |zn − xn| → 0 and hence |(zn) − (xn)|∈E. Therefore (xn) =(zn) .
Denote by C∗ the set of all (u)∗-Cauchy sequences in A∗.
∗ ∗ Theorem 3.10 Let (xn) be a sequence in A. Then (xn) ∈ C if and only if ((xn) ) ∈ C .
Proof. Let (xn) ∈ C. Then for each k ∈ N there exists nk ∈ N such that k|xm − xn|≤u ∗ ∗ for each m, n ∈ N,m,n≥ nk. Thus (u) ≥ (k|xm − xn|) and hence (u) ≥ (k|xm − xn|) = ∗ ∗ ∗ ∗ k|(xm) − (xn) | for each m, n ∈ N,m,n≥ nk. Therefore ((xn) ) ∈ C . ∗ ∗ ∗ ∗ Let ((xn) ) ∈ C . Then for each k ∈ N there exists nk ∈ N such that k|(xm) − (xn) | = ∗ ∗ |(k(xm − xn)) |≤(u) for each m, n ∈ N,m,n≥ nk. ∗ From Lemma 3.5 it follows that (k|xm − xn|) ≤ (u)+(tl), where (tl) ∈ (0) , (0) ≤ (tl). u Hence tl → 0. Thus for each p ∈ N there exists np ∈ N, such that p|tl|≤u for each l ∈ N, l ≥ np. For k =1wehavetl ≤ u for each l ∈ N,l≥ n1.
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Hence k|xm − xn|≤2u, for each m, n ∈ N,m,n≥ nk0 =max{nk,n1}. Then for any k ∈ N there exists nk1 ∈ N such that 2k|xm − xn|≤2u for each m, n ∈ N,m,n≥ nk1 . This implies that for each k ∈ N there exists nk1 ∈ N such that k|xm −xn|≤u for each m, n ∈ N,m,n≥ nk1 . Therefore (xn) ∈ C.
∗ ∗ (u) ∗ Theorem 3.11 Let (xn) ∈ C. Then (xn) → (xn) .
Proof. If (xn) ∈ C, then for each k ∈ N there exists nk ∈ N such that u ≥ k|xm − xn| for each m, n ∈ N,m,n≥ n . n n , u ≥ k|x − x | m ∈ N,m≥ n . k If we take = k we get m nk for each k n n ,n n u ≥ k|x −x |,u≥ k|x −x |, Similarly, if we take = k +1 = k +2,...,wecanget m nk+1 m nk+2 m ∈ N,m≥ n . u ∗ ≥ k|x − x |,k|x − x |,k|x − x |,... ∗. ..., foreach k Hence ( ) ( m nk m nk+1 m nk+2 ) In u ∗ ≥ k|x −x |,k|x −x |,...,k|x −x |,k|x −x |,... ∗ view of Lemma 3.9 we obtain ( ) ( m 1 m 2 m nk m nk+1 ) = ∗ ∗ ∗ ∗ ∗ (u) ∗ (k|(xm) − (xn)|) = k|(xm) − (xn) | for each m ∈ N,m≥ nk. Therefore (xn) → (xn) .
Theorem 3.12 Let ϕ : A → A∗ be the mapping such that ϕ(x)=(x)∗ for each x ∈ A. Then (i) ϕ is a monomorphism, (ii) every element of A∗ is the (u)∗-limit of some sequence in ϕ(A).
Proof. (i) Let x, y ∈ A. Clearly ϕ(x + y)=ϕ(x)+ϕ(y),ϕ(x − y)=ϕ(x) − ϕ(y),ϕ(x ∧ y)= ϕ(x) ∧ ϕ(y),ϕ(x ∨ y)=ϕ(x) ∨ ϕ(y). If ϕ(x)=ϕ(y), then (x)∗ =(y)∗. This implies (x) ∈ (y)∗. Then |(x) − (y)| =(|x − y|) ∈ E. Since 0 is the u−limit of the sequence (|x − y|), we have |x − y| =0. Hence x = y. ∗ ∗ ∗ (ii) Let (xn) ∈ A . Then ((xn) ) is a sequence in ϕ(A). Since (xn) ∈ C, from Theorem 3.11 it ∗ ∗ (u) ∗ follows that (xn) → (xn) .
Definition 3.13 Let B be a DRl-semigroup. If every u−Cauchy sequence (xn) in B is u−con- vergent, then B is called u−Cauchy complete.
Definition 3.14 Let B be a DRl-semigroup. A DRl-semigroup D is said to be a u−Cauchy completion of B, if the following conditions are satisfied: (i) B is a sub-DRl-semigroup of D, (ii) D is u−Cauchy complete, (iii) Every element of D is a u−limit of some sequence in B.
Theorem 3.15 A∗ is a (u)∗−Cauchy complete DRl-semigroup.
1 1 ∗ 2 2 ∗ ∗ n n ∗ Proof. Let X =(xm) , X =(xm) ,... be a sequence in C . Let n ∈ N. Let Xm =(xm) n for each m ∈ N. Hence Xm ∈ ϕ(A) for each m ∈ N. By Theorem 3.11, the sequence n n n ∗ n (X1 ,X2 ,X3 ,...)(u) −converges to X . Hence for choosen n ∈ N there exists mn ∈ N, such n n ∗ that n|Xm − X |≤(u) for each m ∈ N,m≥ mn. If we let run n over N, we can take m ≤ m ≤··· . m m , n|Xn − Xn|≤ u ∗ n ∈ N. 1 2 Further, if we take = n we get mn ( ) for each Let Z Xn n ∈ N. n|Z − Xn|≤ u ∗ n ∈ N. n = mn for each Then n ( ) for any
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∗ n ∗ Now we show that (Zn) ∈ C . Since (X )isan(u) −Cauchy sequence, for each l ∈ N m n ∗ there exists nl ∈ N,nl >lsuch that l|X − X |≤(u) for each m, n ∈ N,m,n≥ nl. Since m m ∗ n n n l ≤ m, n, we get l|Zm − X |≤m|Zm − X |≤(u) ,l|X − Zn|≤n|X − Zn| = n|Zn − X |≤ ∗ ∗ m m n (u) for each m, n ∈ N,m,n≥ nl. Then we have 2(u) ≥ l|Zm − X | + l|X − X | = m m n n l(|Zm − X | + |X − X |) ≥ l(|Zm − X |) for each m, n ∈ N,m,n≥ nl. Hence for any ∗ n l ∈ N there exists nl1 ∈ N,nl1 ≥ nl such that 2(u) ≥ 2l|Zm − X | for each m, n ∈ N, ∗ n m, n ≥ nl1 . Therefore (u) ≥ l|Zm − X | for each m, n ∈ N,m,n≥ nl1 . Further, we obtain ∗ n n n n 2(u) ≥ l|Zm − X | + l|X − Zn| = l(|Zm − X | + |X − Zn|) ≥ l|Zm − Zn| for each m, n ∈ N, ∗ m, n ≥ nl1 . Thus for any l ∈ N there exists nl2 ∈ N,nl2 ≥ nl1 such that 2(u) ≥ 2l|Zm − Zn| for each m, n ∈ N,m,n≥ nl2 . Therefore for each l ∈ N there exists nl2 ∈ N such that ∗ ∗ (u) ≥ l|Zm − Zn| for each m, n ∈ N,m,n≥ nl2 . Therefore (Zn) ∈ C . z xn , Z z ∗. Z ∈ C∗, If we put n = mn then n =( n) Since ( n) from Theorem 3.10 it follows that ∗ (zn) ∈ C. By Theorem 3.11, Zn → (zn) . ∗ ∗ ∗ Let t ∈ N. Since Zn → (zn) , there exist nt ∈ N, nt ≥ t such that t|Zk − (zn) |≤(u) for each k ∈ N,k≥ nt. k k k ∗ Since t ≤ k, we have t|X − Zk| = t|Zk − X |≤k|Zk − X |≤(u) for each k ∈ N,k≥ nt. ∗ k ∗ k ∗ k ∗ Then we have 2(u) ≥ t|X − Zk| + t|Zk − (zn) | = t(|X − Zk| + |Zk − (zn) |) ≥ t|X − (zn) | ∗ for each k ∈ N,k≥ nt. Thus for any t ∈ N there exists nt1 ∈ N, nt1 ≥ nt such that 2(u) ≥ k ∗ 2t|X − (zn) | for each k ∈ N,k≥ nt1 . Hence for each t ∈ N there exists nt1 ∈ N such that ∗ ∗ k ∗ n (u) ∗ (u) ≥ t|X − (zn) | for each k ∈ N,k≥ nt. Therefore X −→ (zn) . If x and ϕ(x)=(x)∗ will be identified for each x ∈ A, then A is a subgroup of A∗ and (u)∗ = u. Then we get as a consequence of Theorems 3.12 and 3.15 the following proposition.
Theorem 3.16 A∗ is an u−Cauchy completion of A.
Theorem 3.17 If A1 and A2 are u−Cauchy completions of A, then there exists a semigroup l-isomorphism of A1 onto A2 leaving all elements of A fixed.
Proof. From the assumptions it follows that A is a sub-DRl-semigroup of A1 and A2. Let u x ∈ A1. Then there exists a sequence (xn)inAsuchthatxn → x in A1. Since (xn)is a convergent, (xn)isanu−Cauchy sequence in A1 andthenalsoinA and A2. Since A2 is u u−Cauchy complete, there exists x ∈ A2 such that xn → x in A2. We put ψ(x )=x . Now we show that the mapping ψ is correctly defined. Let (yn) be another sequence in A u such that yn → x in A1. Analogously as above we can get that there exists x ∈ A2 such u u u u that yn → x in A2. Then we have xn − yn → 0,yn − xn → 0inA1 and xn − yn → x − x , u u u yn − xn → x − x in A2. Since 0 ∈ A ⊆ A2, we have xn − yn → 0inA2,yn − xn → 0in A2, Because u-limits are uniquely determined we obtain x − x =0,x − x =0.Then(P1) yields x ≤ x x ≤ x . Therefore x = x . u Let z ∈ A2. Then there exists a sequence (zn) ∈ A such that zn → z in A2 and z ∈ A1 u such that zn → z in A1. Then ψ(z )=z . Therefore ψ is a surjective mapping. Let a ,b ∈ A1, u u ψ(a )=a , ψ(b )=b . Hence there are sequences (an)and(bn)inA such that an → a , bn → b u u u u u in A1 and an → a , bn → b in A2. Then an + bn → a + b ,an − bn → a − b ,an ∨ bn → a ∨ b , u u u u u an ∧bn → a ∧b in A1, an +bn → a +b ,an −bn → a −b ,an ∨bn → a ∨b ,an ∧bn → a ∧b in A2. Therefore ψ(a + b )=a + b = ψ(a )+ψ(b ),ψ(a − b )=a − b = ψ(a ) − ψ(b ), ψ(a ∨ b )=a ∨ b = ψ(a ) ∨ ψ(b ),ψ(a ∧ b )=a ∧ b = ψ(a ) ∧ ψ(b ).
volume 5 (2012), number 3 21 Aplimat - Journal of Applied Mathematics